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Über dieses Buch

The topics in this research monograph are at the interface of several areas of mathematics such as harmonic analysis, functional analysis, analysis on spaces of homogeneous type, topology, and quasi-metric geometry. The presentation is self-contained with complete, detailed proofs, and a large number of examples and counterexamples are provided.

Unique features of Metrization Theory for Groupoids: With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis include:

* treatment of metrization from a wide, interdisciplinary perspective, with accompanying applications ranging across diverse fields;

* coverage of topics applicable to a variety of scientific areas within pure mathematics;

* useful techniques and extensive reference material;

* includes sharp results in the field of metrization.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

* coverage of topics applicable to a variety of scientific areas within pure mathematics;

* useful techniques and extensive reference material;

* includes sharp results in the field of metrization.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

* useful techniques and extensive reference material;

* includes sharp results in the field of metrization.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

* includes sharp results in the field of metrization.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
In this chapter we state a general metrization theorem, in the algebraic setting of groupoids, and explain its relationship to several classical results in analysis such as the Macías–Segovia metrization theorem for quasimetric spaces, the Aoki–Rolewicz theorem for quasinormed vector spaces, and the Alexandroff–Urysohn metrization theorem for uniform spaces. The metrization theorem in question is quantitative in nature and involves starting from a given quasisubadditive function defined on the underlying groupoid. We also indicate that our general metrization theorem is sharp.
Dorina Mitrea, Irina Mitrea, Marius Mitrea, Sylvie Monniaux

Chapter 2. Semigroupoids and Groupoids

Abstract
This chapter amounts to a concise, self-contained introduction to the theory of semigroupoids and groupoids, from an algebraic and topologic point of view. In particular, a multitude of examples are presented and analyzed. On the algebraic side, an alternative description of Brant groupoids is provided and a structure theorem for semigroupoids established. On the topological side, the notion of topological groupoid is introduced and studied. To set the stage for future metrization results, the concept of partially defined distance is also considered here.
Dorina Mitrea, Irina Mitrea, Marius Mitrea, Sylvie Monniaux

Chapter 3. Quantitative Metrization Theory

Abstract
This chapter contains our most general abstract results pertaining to the metrization of semigroupoids and groupoids equipped with quasisubadditive functions. Moreover, several metrization results in this setting with additional constraints are established. We explain how our results generalize the classic Aoki–Rolewicz theorem for quasinormed vector spaces and yield a sharp version of the Macías–Segovia metrization theorem for quasimetric spaces. An application to analytic capacity is also discussed, and connections with homogeneous groups are explored.
Dorina Mitrea, Irina Mitrea, Marius Mitrea, Sylvie Monniaux

Chapter 4. Applications to Analysis on Quasimetric Spaces

Abstract
In this chapter we study the implications of our general metrization theory at the level of quasimetric spaces, with special emphasis on analytical aspects. More specifically, we study the nature of Hölder functions on quasimetric spaces by proving density, embeddings, separation, and extension theorems. We also quantify the richness of such spaces by introducing and studying a notion of index that interfaces tightly with the critical exponent beyond which the Hölder spaces become trivial. Other applications are targeted to Hardy spaces on spaces of homogeneous type, regularized distance, Whitney decompositions, and partitions of unity, as well as the Gromov–Pompeiu–Hausdorff distance.
Dorina Mitrea, Irina Mitrea, Marius Mitrea, Sylvie Monniaux

Chapter 5. Nonlocally Convex Functional Analysis

Abstract
In this chapter the goal is to explore the implications of our general metrization theory to aspects of functional analysis in nonlocally convex topological vector spaces. Some of the concrete topics studied here deal with the completeness and separability of such spaces, as well as with the issues of pointwise convergence and the Fatou property in the case of function spaces. Various other applications to analysis on Boolean algebras are also considered.
Dorina Mitrea, Irina Mitrea, Marius Mitrea, Sylvie Monniaux

Chapter 6. Functional Analysis on Quasi-Pseudonormed Groups

Abstract
The aim of this chapter is to explore the ramifications of our general metrization theory for classic functional analysis concerned with open mapping theorems, closed graph theorems, and uniform boundedness principles, for which we establish a new generation of results. Here we also prove a refinement of the classic Birkhoff–Kakutani theorem by fully bringing the topology into focus. The prevalent setting in most of the results established in this chapter is that of quasi-pseudonormed group.
Dorina Mitrea, Irina Mitrea, Marius Mitrea, Sylvie Monniaux

Backmatter

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